This paper proposes a novel approach to integrating partial differential equation (PDE)-based evolution models into neural networks through a new type of regularization. Specifically, we propose inverse evolution layers (IELs) based on evolution equations. These layers can achieve specific regularization objectives and endow neural networks' outputs with corresponding properties of the evolution models. Moreover, IELs are straightforward to construct and implement, and can be easily designed for various physical evolutions and neural networks. Additionally, the design process for these layers can provide neural networks with intuitive and mathematical interpretability, thus enhancing the transparency and explainability of the approach. To demonstrate the effectiveness, efficiency, and simplicity of our approach, we present an example of endowing semantic segmentation models with the smoothness property based on the heat diffusion model. To achieve this goal, we design heat-diffusion IELs and apply them to address the challenge of semantic segmentation with noisy labels. The experimental results demonstrate that the heat-diffusion IELs can effectively mitigate the overfitting problem caused by noisy labels.

本文提出一种新方法，通过一种新类型的正则化将基于偏微分方程（PDE）的演化模型整合到神经网络中，这些反演演化层（IEL）可以实现特定的正则化目标并赋予神经网络输出相应的演化模型属性，进而提高透明度和可解释性。

反演进化层: 深度神经网络的物理学知识正则化器